you should not have to wash your eggs. chickens take great care with their eggs. if they are pooping on the eggs then they probably have worms. and the treatment for that is simple. you should do this to your chickens once a month. give them diatomaceous earth in their coop and in their food
There is a lot of debate on whether DE actually treats worms in chickens.
It is a matter of preference for me.
I prefer to have a fecal test done and then treat if needed with a known effective product.